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arXiv · 2211.11118

The Internal Operads of Combinatory Algebras

Masahito Hasegawa
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Abstract

We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar $\mathbf{BI}(\_)^\bullet$-algebras as well as the braided $\mathbf{BC^\pm I}$-algebras. We show that every extensional combinatory algebra gives rise to a canonical closed operad, which we shall call the internal operad of the combinatory algebra. The internal operad construction gives a left adjoint to the forgetful functor from closed operads to extensional combinatory algebras. As a by-product, we derive extensionality axioms for the classes of combinatory algebras mentioned above.

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Masahito Hasegawa. 2022-11-20. The Internal Operads of Combinatory Algebras. https://doi.org/10.46298/entics.10338

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